Round answers according to the rounding conventions on page 364. Solve the triangles. C=114.2^∘,

Round answers according to the rounding conventions on page...

Key Concepts

Angle Sum Property

The Angle Sum Property of a triangle states that the sum of its interior angles is always 180 degrees. This principle is crucial when solving triangles, as it allows for the determination of a missing angle when the other two angles are already known.

Rounding Conventions

Rounding conventions dictate how to limit the number of significant digits in a numerical result. This concept is essential in ensuring that answers are presented consistently and with the intended precision, often based on guidelines or specific instructions provided in a problem statement.

Law of Sines

The Law of Sines establishes a relationship between the sides and angles of a triangle by stating that the ratio of a side's length to the sine of its opposite angle is the same for all three sides. This concept is fundamental for solving triangles, particularly in cases where an angle and its opposite side are known, enabling the computation of unknown sides or angles.

Transcript

00:01 Given this information about a triangle, right, given any sort of triangle with angles, capital a, b, and c and sides across from the angles, lower case a, b, and c, what i'm given is an angle and two sides.

00:16 So i'm actually given two sides and an angle.

00:20 Okay.

00:21 So here what i notice is that if i'm given two sides and an angle not between the sides, then i actually have three different cases here to consider.

00:29 I can have zero triangles, one triangle, or i could have two triangles.

00:34 First thing i need to know is the angle measure.

00:36 Is my angle acute or obtuse? well, it's greater than 90 degrees, so it's going to be obtuse.

00:44 Okay, so this eliminates my two triangles.

00:47 I can either have zero or one triangles or one triangle.

00:52 Now what i have to look at, i have to look at c and b in that relationship there.

00:57 So i see that c is 87.

00:58 And b is 12.

01:00 So c is actually greater than b.

01:02 So this means that i will have exactly one triangle.

01:06 If c was less than b, then i would have no triangles that meet this criteria.

01:11 Okay, so i have one triangle that i'm trying to solve for.

01:14 Which side or angle i'm going to solve for next? well, i know nothing about the a's.

01:20 I know all the c's, so but i need to find angle b.

01:23 Am i going to use law of signs or law of cosons? well, i'm going to use law of signs because for law of cousins, i need at least, if i'm solving for a side, i need at least two sides.

01:38 But if i'm solving for an angle, i need all three sides.

01:40 So law of signs, sign of angle b over side length b, 12 .1 equals sign of angle c, 114 .2 over sideline c, 87 .2.

01:54 I'm trying to get sign of b by itself.

01:56 So i'm going to multiply my 12 .1 to both sides...

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